Integers - PSAT Math
Card 1 of 1267
If 
is a positive number, and 
is also a positive number, what is a possible value for 
?
If
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Because -3b is positive, b must be negative since the product of two negative numbers is positive.
Because ab is also positive, a must also be negative in order to produce a prositive product.
To check you answer, you can try plugging in any negative number for a.
Because -3b is positive, b must be negative since the product of two negative numbers is positive.
Because ab is also positive, a must also be negative in order to produce a prositive product.
To check you answer, you can try plugging in any negative number for a.
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Find the product.
Find the product.
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When multiplying together two negatives, our value for the product become positive. 
When multiplying together two negatives, our value for the product become positive.
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Find the product.
Find the product.
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Since we have one positive and one negative multiple, the resulting product must be negative. 
Since we have one positive and one negative multiple, the resulting product must be negative.
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Let 
be a negative integer and 
be a nonzero integer. Which of the following must be negative regardless of whether 
is positive or negative?
Let
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Since 
is positive, 
, the product of a negative number and a positive number, must be negative also.
Of the others:
is incorrect; if 
is negative, then 
is positive, and 
assumes the sign of 
.
is incorrect; again, 
is positive, and if 
is a positive number, 
is positive.
is incorrect; regardless of the sign of 
, 
is positive, and if its absolute value is greater than that of 
, 
is positive.
Since
Of the others:
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Given that 
are both integers, 
, and 
, which of the following is correct about the sign of the expression 
?
Given that
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If 
, then we know that 
is any number between or equal to 
and 
. Therefore 
must be a negative number.
Also, if 
, then we know that 
is any number between or equal to 
and 
. Therefore 
must be a negative number.
Now looking the expression 
we can find the sign of each component in the expression.
Since 
is negative, we know that a negative number minus another number is still a negative number.
Therefore, 
is a negative number.
Since 
is between or equal to 
and 
we can plug in these end values in to determine the sign of 
.
Therefore, 
is either zero or a positive number.
Now to find the sign of the expression we look at the product of the two components. The product of a negative number and a positive number is a negative number; the product of a negative number and zero is zero. Therefore, the correct choice is that 
is negative or zero.
If
Also, if
Now looking the expression
Since
Therefore,
Since
Therefore,
Now to find the sign of the expression we look at the product of the two components. The product of a negative number and a positive number is a negative number; the product of a negative number and zero is zero. Therefore, the correct choice is that
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and 
are positive numbers; 
is a negative number. All of the following must be positive except:
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Since 
and 
are positive, all powers of 
and 
will be positive; also, in each of the expressions, the powers of 
and 
are being added. The clue to look for is the power of 
and the sign before it.
In the cases of 
and 
, since the negative number 
is being raised to an even power, each expression amounts to the sum of three positive numbers, which is positive.
In the cases of 
and 
, since the negative number 
is being raised to an odd power, the middle power is negative - but since it is being subtracted, it is the same as if a positive number is being added. Therefore, each is essentially the sum of three positive numbers, which, again, is positive.
In the case of 
, however, since the negative number 
is being raised to an odd power, the middle power is again negative. This time, it is basically the same as subtracting a positive number. As can be seen in this example, it is possible to have this be equal to a negative number:
:
Therefore, 
is the correct choice.
Since
In the cases of
In the cases of
In the case of
Therefore,
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, 
, and 
are all negative numbers. Which of the following must be positive?
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The key is knowing that a negative number raised to an odd power yields a negative result, and that a negative number raised to an even power yields a positive result.
: 
and 
are positive, yielding a positive dividend; 
is a negative divisor; this result is negative.
: 
and 
are negative, yielding a positive dividend; 
is a negative divisor; this result is negative.
: 
is positive and 
is negative, yielding a negative dividend; 
is a positive divisor; this result is negative.
: 
is negative and 
is positive, yielding a negative dividend; 
is a positive divisor; this result is negative.
: 
is positive and 
is negative, yielding a negative dividend; 
is a negative divisor; this result is positive.
The correct choice is 
.
The key is knowing that a negative number raised to an odd power yields a negative result, and that a negative number raised to an even power yields a positive result.
The correct choice is
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is a positive integer; 
and 
are negative integers. Which of the following three expressions must be negative?
I) 
II) 
III) 
I)
II)
III)
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A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even; it is negative if and only if the absolute value iof the exponent is odd. Therefore, all three expressions have signs that are dependent on the odd/even parity of 
and 
, which are not given in the problem.
The correct response is none of these.
A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even; it is negative if and only if the absolute value iof the exponent is odd. Therefore, all three expressions have signs that are dependent on the odd/even parity of
The correct response is none of these.
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, 
, and 
are all negative odd integers. Which of the following three expressions must be positive?
I) 
II) 
III) 
I)
II)
III)
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A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even. Since the sum or difference of two odd integers is always an even integer, this is the case in all three expressions. The correct response is all of these.
A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even. Since the sum or difference of two odd integers is always an even integer, this is the case in all three expressions. The correct response is all of these.
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If is a positive number, and is also a positive number, what is a possible value for ?
If
Tap to reveal answer
Because -3b is positive, b must be negative since the product of two negative numbers is positive.
Because ab is also positive, a must also be negative in order to produce a prositive product.
To check you answer, you can try plugging in any negative number for a.
Because -3b is positive, b must be negative since the product of two negative numbers is positive.
Because ab is also positive, a must also be negative in order to produce a prositive product.
To check you answer, you can try plugging in any negative number for a.
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Find the product.
Find the product.
Tap to reveal answer
When multiplying together two negatives, our value for the product become positive.
When multiplying together two negatives, our value for the product become positive.
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Find the product.
Find the product.
Tap to reveal answer
Since we have one positive and one negative multiple, the resulting product must be negative.
Since we have one positive and one negative multiple, the resulting product must be negative.
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Let be a negative integer and be a nonzero integer. Which of the following must be negative regardless of whether is positive or negative?
Let
Tap to reveal answer
Since is positive, , the product of a negative number and a positive number, must be negative also.
Of the others:
is incorrect; if is negative, then is positive, and assumes the sign of .
is incorrect; again, is positive, and if is a positive number, is positive.
is incorrect; regardless of the sign of , is positive, and if its absolute value is greater than that of , is positive.
Since
Of the others:
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Given that are both integers, , and , which of the following is correct about the sign of the expression ?
Given that
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If , then we know that is any number between or equal to and . Therefore must be a negative number.
Also, if , then we know that is any number between or equal to and . Therefore must be a negative number.
Now looking the expression we can find the sign of each component in the expression.
Since is negative, we know that a negative number minus another number is still a negative number.
Therefore, is a negative number.
Since is between or equal to and we can plug in these end values in to determine the sign of .
Therefore, is either zero or a positive number.
Now to find the sign of the expression we look at the product of the two components. The product of a negative number and a positive number is a negative number; the product of a negative number and zero is zero. Therefore, the correct choice is that is negative or zero.
If
Also, if
Now looking the expression
Since
Therefore,
Since
Therefore,
Now to find the sign of the expression we look at the product of the two components. The product of a negative number and a positive number is a negative number; the product of a negative number and zero is zero. Therefore, the correct choice is that
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and are positive numbers; is a negative number. All of the following must be positive except:
Tap to reveal answer
Since and are positive, all powers of and will be positive; also, in each of the expressions, the powers of and are being added. The clue to look for is the power of and the sign before it.
In the cases of and , since the negative number is being raised to an even power, each expression amounts to the sum of three positive numbers, which is positive.
In the cases of and , since the negative number is being raised to an odd power, the middle power is negative - but since it is being subtracted, it is the same as if a positive number is being added. Therefore, each is essentially the sum of three positive numbers, which, again, is positive.
In the case of , however, since the negative number is being raised to an odd power, the middle power is again negative. This time, it is basically the same as subtracting a positive number. As can be seen in this example, it is possible to have this be equal to a negative number:
:
Therefore, is the correct choice.
Since
In the cases of
In the cases of
In the case of
Therefore,
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, , and are all negative numbers. Which of the following must be positive?
Tap to reveal answer
The key is knowing that a negative number raised to an odd power yields a negative result, and that a negative number raised to an even power yields a positive result.
: and are positive, yielding a positive dividend; is a negative divisor; this result is negative.
: and are negative, yielding a positive dividend; is a negative divisor; this result is negative.
: is positive and is negative, yielding a negative dividend; is a positive divisor; this result is negative.
: is negative and is positive, yielding a negative dividend; is a positive divisor; this result is negative.
: is positive and is negative, yielding a negative dividend; is a negative divisor; this result is positive.
The correct choice is .
The key is knowing that a negative number raised to an odd power yields a negative result, and that a negative number raised to an even power yields a positive result.
The correct choice is
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is a positive integer; and are negative integers. Which of the following three expressions must be negative?
I)
II)
III)
I)
II)
III)
Tap to reveal answer
A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even; it is negative if and only if the absolute value iof the exponent is odd. Therefore, all three expressions have signs that are dependent on the odd/even parity of and , which are not given in the problem.
The correct response is none of these.
A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even; it is negative if and only if the absolute value iof the exponent is odd. Therefore, all three expressions have signs that are dependent on the odd/even parity of
The correct response is none of these.
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, , and are all negative odd integers. Which of the following three expressions must be positive?
I)
II)
III)
I)
II)
III)
Tap to reveal answer
A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even. Since the sum or difference of two odd integers is always an even integer, this is the case in all three expressions. The correct response is all of these.
A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even. Since the sum or difference of two odd integers is always an even integer, this is the case in all three expressions. The correct response is all of these.
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How many prime numbers are between 1 to 25?
How many prime numbers are between 1 to 25?
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A prime numbers is a number greater than 1 that can only be divided by 1 and itself. Simply count from 1 to 25 and see how many values fit the criteria.
2, 3, 4, 5, 6, 7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
Prime numbers are underlined. Nine prime numbers are in this set interval.
A prime numbers is a number greater than 1 that can only be divided by 1 and itself. Simply count from 1 to 25 and see how many values fit the criteria.
2, 3, 4, 5, 6, 7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
Prime numbers are underlined. Nine prime numbers are in this set interval.
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Four consecutive odd integers sum to 40. How many of these numbers are prime?
Four consecutive odd integers sum to 40. How many of these numbers are prime?
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Let x equal the smallest of the four numbers. Therefore:
x + (x+2)+(x+4)+(x+6)=40
4x +12 = 40
4x + 28
x=7
Therefore the four odd numbers are 7, 9, 11, and 13. Since all are prime except 9, three of the numbers are prime.
Let x equal the smallest of the four numbers. Therefore:
x + (x+2)+(x+4)+(x+6)=40
4x +12 = 40
4x + 28
x=7
Therefore the four odd numbers are 7, 9, 11, and 13. Since all are prime except 9, three of the numbers are prime.
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